Past exam of the approximation theory and finite element methods course of the University of Oxford / 2010 / A / 3 / a / Solution

Put $m={\min }_r|u(x_r)-q(x_r)|$ and $M={\max }_r|u(x_r)-q(x_r)|$. If the best uniform approximation had error below $m$, the values of $p_n^{\ast }-q$ would strictly alternate in sign at all $n+2$ reference points. The intermediate value theorem would give at least $n+1$ distinct roots of a polynomial of degree at most $n$, a contradiction. If $m=0$ the lower bound is immediate.

At least one reference point attains the global error maximum of $q$, so $M=\Vert u-q{\Vert }_{\infty }$. Optimality of $p_n^{\ast }$ gives the upper bound, and thereforeThis is the unequal-height version of the alternating-reference error bound.